The cost formulation (REF ) suggests an alternating minimization algorithm, which alternatively updates \(Q\)  and \(\mathbb {Y}\) . For a given \(\mathbb {Y}\) , \(Q\)  is the matrix of singular vectors corresponding to the tail singular values. For a given \(Q\) , updating \(\mathbb {Y}\)  is a regularized least squares problem. One method for this regularized least-squares sub-problem is proximal gradient descent [1]}, [2]}, where for a function \(\mathcal {F}(\mathbb {Y})\) , its proximal operator is \(\text{prox}_\mathcal {F}(\mathbb {Y}) = \arg \min _\mathcal {\mathbb {X}} \frac{1}{2}\Vert \mathbb {X}-\mathbb {Y}\Vert _F^2+\mathcal {F}(\mathbb {X})\) . Proximal gradient descent for updating \(\mathbb {Y}\)  given \(Q^{(k)}\)  is
\(\mathbb {Y}^{(k+1)} = \text{prox}_{\eta _k\mathcal {E}} \left( \mathbb {Y}^{(k)}-\eta _k\nabla _{\mathbb {Y}} \left\Vert \mathcal {H}_\mathbb {K}(\mathbb {Y})Q^{(k)} \right\Vert _F^2\bigg |_{\mathbb {Y}=\mathbb {Y}^{(k)}} \right)\) 
